18edf: Difference between revisions
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== Theory == | |||
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]]. | |||
Lookalikes: [[31edo]], [[49edt]], [[72ed5]], [[80ed6]] | Lookalikes: [[31edo]], [[49edt]], [[72ed5]], [[80ed6]] | ||
==Harmonics== | === Harmonics === | ||
{{Harmonics in equal|18|3|2}} | {{Harmonics in equal|18|3|2}} | ||
{{Harmonics in equal|18|3|2|start=12|collapsed=1}} | {{Harmonics in equal|18|3|2|start=12|collapsed=1}} | ||
==Intervals== | == Intervals == | ||
{| class="wikitable mw-collapsible" | {| class="wikitable mw-collapsible" | ||
|+ Intervals of 18edf | |+ Intervals of 18edf | ||
| Line 204: | Line 205: | ||
|} | |} | ||
==Related regular temperaments== | == Related regular temperaments == | ||
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator. | The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator. | ||
===7-limit 31&369=== | === 7-limit 31&369 === | ||
Commas: 2401/2400, 8589934592/8544921875 | Commas: 2401/2400, 8589934592/8544921875 | ||
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EDOs: 31, 369, 400, 431, 462 | EDOs: 31, 369, 400, 431, 462 | ||
{{ | {{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}} | ||
Revision as of 16:56, 17 January 2025
| ← 17edf | 18edf | 19edf → |
18 equal divisions of the perfect fifth (abbreviated 18edf or 18ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 18 equal parts of about 39 ¢ each. Each step represents a frequency ratio of (3/2)1/18, or the 18th root of 3/2.
Theory
18edf is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; with 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.
Lookalikes: 31edo, 49edt, 72ed5, 80ed6
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.9 | +8.9 | +17.8 | -17.5 | +17.8 | -15.0 | -12.2 | +17.8 | -8.6 | -17.6 | -12.2 |
| Relative (%) | +22.9 | +22.9 | +45.8 | -44.9 | +45.8 | -38.6 | -31.4 | +45.8 | -22.0 | -45.1 | -31.4 | |
| Steps (reduced) |
31 (13) |
49 (13) |
62 (8) |
71 (17) |
80 (8) |
86 (14) |
92 (2) |
98 (8) |
102 (12) |
106 (16) |
110 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.2 | -6.1 | -8.6 | -3.3 | +8.7 | -12.2 | +11.2 | +0.4 | -6.1 | -8.7 | -7.6 |
| Relative (%) | +13.3 | -15.7 | -22.0 | -8.5 | +22.4 | -31.4 | +28.6 | +0.9 | -15.7 | -22.2 | -19.5 | |
| Steps (reduced) |
114 (6) |
117 (9) |
120 (12) |
123 (15) |
126 (0) |
128 (2) |
131 (5) |
133 (7) |
135 (9) |
137 (11) |
139 (13) | |
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | exact 1/1 | ||
| 1 | 38.9975 | 45/44 | |
| 2 | 77.995 | ||
| 3 | 116.9925 | 16/15 | |
| 4 | 155.99 | 128/117 | |
| 5 | 194.9875 | 28/25 | |
| 6 | 233.985 | 8/7 | |
| 7 | 272.9825 | 7/6 | |
| 8 | 311.98 | 6/5 | |
| 9 | 350.9775 | 60/49, 49/40 | |
| 10 | 389.975 | 5/4 | |
| 11 | 428.9725 | 9/7 | |
| 12 | 467.97 | ||
| 13 | 506.9675 | 75/56 | |
| 14 | 545.965 | ||
| 15 | 584.9625 | ||
| 16 | 623.96 | ||
| 17 | 662.9575 | 22/15 | |
| 18 | 701.955 | exact 3/2 | just perfect fifth |
| 19 | 740.9525 | 135/88 | |
| 20 | 779.95 | ||
| 21 | 818.9475 | 8/5 | |
| 22 | 857.945 | 64/39 | |
| 23 | 896.9425 | 42/25 | |
| 24 | 935.94 | 12/7 | |
| 25 | 974.9375 | 7/4 | |
| 26 | 1013.935 | 9/5 | |
| 27 | 1052.9325 | 90/49, 147/80 | |
| 28 | 1091.93 | 15/8 | |
| 29 | 1130.9275 | 27/14 | |
| 30 | 1169.925 | ||
| 31 | 1208.9225 | 225/112 | |
| 32 | 1247.92 | ||
| 33 | 1286.9175 | ||
| 34 | 1325.915 | ||
| 35 | 1364.9125 | ||
| 36 | 1403.91 | exact 9/4 | |
Related regular temperaments
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
7-limit 31&369
Commas: 2401/2400, 8589934592/8544921875
POTE generator: ~5/4 = 386.997
Mapping: [<1 19 2 7|, <0 -54 1 -13|]
11-limit 31&369
Commas: 2401/2400, 5632/5625, 46656/46585
POTE generator: ~5/4 = 386.999
Mapping: [<1 19 2 7 37|, <0 -54 1 -13 -104|]
EDOs: 31, 369, 400, 431, 462
13-limit 31&369
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
POTE generator: ~5/4 = 387.003
Mapping: [<1 19 2 7 37 -35|, <0 -54 1 -13 -104 120|]
EDOs: 31, 369, 400, 431, 462
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Todo: cleanup , expand say what the temperaments are like and why one would want to use them, and for what |